Introduction

According to classical physics (i.e., physics prior to the 20th century), particles and waves are distinct classes of physical entity that possess markedly different properties. For instance, particles are discrete, which means that they cannot be arbitrarily divided. In other words, it makes sense to talk about one electron, or two electrons, but not about a third of an electron. Waves, on the other hand, are continuous, which means that they can be arbitrarily divided. In other words, given a wave whose amplitude has a certain value, it makes sense to talk about a similar wave whose amplitude is one third, or any other fraction whatsoever, of this value. Particles are also highly localized in space. For example, atomic nuclei have very small radii of order $10^{-15} {\rm m}$ , whereas electrons act like point particles (i.e., they have no discernible spatial extent). Waves, on the other hand, are non-localized in space. In fact, a wave is defined as a disturbance that is periodic in space, with some finite periodicity length (i.e., wavelength). Hence, it is fairly meaningless to talk about a disturbance being a wave unless it extends over a region of space that is at least a few wavelengths in size.

The classical scenario, just described, in which particles and waves are distinct phenomena, had to be significantly modified in the early decades of the 20th century. During this time period, physicists discovered, much to their surprise, that, under certain circumstances, waves act as particles, and particles act as waves. This bizarre behavior is known as wave-particle duality. For instance, the photoelectric effect (see Section C.2) shows that electromagnetic waves sometimes act like swarms of massless particles called photons. Moreover, the phenomenon of electron diffraction by atomic lattices (see Section C.3) implies that electrons sometimes possess wave-like properties. Wave-particle duality usually only manifests itself on atomic and sub-atomic lengthscales [i.e., on lengthscales less than, or of order, $10^{-10} {\rm m}$ . (See Section C.3).] The classical picture remains valid on significantly longer lengthscales. Thus, on macroscopic lengthscales, waves only act like waves, particles only act like particles, and there is no wave-particle duality. However, on atomic lengthscales, classical mechanics, which governs the macroscopic behavior of massive particles, and classical electrodynamics, which governs the macroscopic behavior of electromagnetic fields--neither of which take wave-particle duality into account--must be replaced by new theories. The theories in question are called quantum mechanics and quantum electrodynamics, respectively. In this appendix, we shall discuss a simple version of quantum mechanics in which the microscopic dynamics of massive particles (i.e., particles with finite mass) is described entirely in terms of wavefunctions. This particular version of quantum mechanics is known as wave mechanics.